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Expressions of sequences for AQA GCSE Maths

Expressions of sequences

This page covers the following topics:

1. nth term of arithmetic sequences
2. nth term of quadratic sequences

To determine the nth term expression for an arithmetic sequence, determine the first term in the sequence and the common first difference. Substituting the values to aโ‚™ = aโ‚ + d(n โˆ’ 1) and simplifying the expression obtained gives the nth term of an arithmetic sequence. To find a specific term in an arithmetic sequence, the value of n can be substituted into the nth term expression.

nth term of arithmetic sequences

To determine the nth term expression for a quadratic sequence, determine the second common difference in the sequence to obtain the coefficient a. Then determine the value b by using one of the first differences and the value of a. Finally, the value of c can be determined by using the expression for any term, preferably the first one. Substituting the values to aโ‚™ = anยฒ + bn + c gives the nth term of a quadratic sequence. To find a specific term in a quadratic sequence, the value of n can be substituted into the nth term expression.

nth term of quadratic sequences

1

Define the nth term of the sequence aโ‚™ = {1, 3, 5, 7, 9, โ€ฆ}.

d = 3 โˆ’ 1 = 2
Using the general formula for the n-th term, aโ‚™ = 1 + 2(n โˆ’ 1) = 2n โˆ’ 1.

2n โˆ’ 1

Define the nth term of the sequence aโ‚™ = {1, 3, 5, 7, 9, โ€ฆ}.

2

Find the nth term, or general rule, of the sequence Uโ‚™ = {โˆ’9, โˆ’6, โˆ’1, 6, 15, โ€ฆ}.

The sequence of first differences is given by {3, 5, 7, 9, ...}.
The common second difference is given by 2.
Using the expressions from the nth term of a quadratic sequence, 2a = 2, thus a = 1.
3 ร— 1 + b = 3, thus b = 0.
1 + 0 + c = โˆ’9, so c = โˆ’10.
Substituting into the general formula for the n-th term, Uโ‚™ = nยฒ โˆ’ 10.

nยฒ โˆ’ 10

Find the nth term, or general rule, of the sequence Uโ‚™ = {โˆ’9, โˆ’6, โˆ’1, 6, 15, โ€ฆ}.

3

Given the common first difference term of an arithmetic sequence is d = โˆ’4 and the second term of the sequence is aโ‚‚ = 17, find the nth term of the sequence.

aโ‚ = 17 โˆ’ (โˆ’4) = 21
Using the general formula for the n-th term, aโ‚™ = 21 + (โˆ’4)(n โˆ’ 1) = 25 โˆ’ 4n.

25 โˆ’ 4n

Given the common first difference term of an arithmetic sequence is d = โˆ’4 and the second term of the sequence is aโ‚‚ = 17, find the nth term of the sequence.

4

Mavis is giving her old clothes to her 4 younger siblings. The number of clothes she hands establishes an arithmetic sequence starting at oldest and running to youngest sibling. The oldest sibling receives 8 pieces of clothing and the youngest receives 2. Find the nth term for the sequence of clothes handed down.

Identify that aโ‚„ = 2 and aโ‚ = 8 and that aโ‚„ โˆ’ aโ‚ = 3d.
2 โˆ’ 8 = โˆ’6 = 3d, therefore d = โˆ’2.
Using the general formula for the n-th term, aโ‚™ = 8 + (โˆ’2)(n โˆ’ 1) = 10 โˆ’ 2n.

10 โˆ’ 2n

Mavis is giving her old clothes to her 4 younger siblings. The number of clothes she hands establishes an arithmetic sequence starting at oldest and running to youngest sibling. The oldest sibling receives 8 pieces of clothing and the youngest receives 2. Find the nth term for the sequence of clothes handed down.

5

Daley works at a charity shop on the weekends starting at 9:00 in the morning. On Saturday he sold 2 shirts in his first hour working, 5 shirts in his second, 10 in his third, and 17 in his fourth. On Sunday he sold twice the number of shirts in each hour. Find the nth term of the sequence of shirts Daley sold on Sunday.

The sequence of first differences is given by {4, 10, 20, 34, ...}.
The common second difference is given by 4.
Using the expressions from the nth term of a quadratic sequence, 2a = 4, thus a = 2.
3 ร— 2 + b = 6, thus b = 0.
2 + 0 + c = 4, so c = 2.
Substituting into the general formula for the n-th term, aโ‚™ = 2nยฒ + 2.

2nยฒ + 2

Daley works at a charity shop on the weekends starting at 9:00 in the morning. On Saturday he sold 2 shirts in his first hour working, 5 shirts in his second, 10 in his third, and 17 in his fourth. On Sunday he sold twice the number of shirts in each hour. Find the nth term of the sequence of shirts Daley sold on Sunday.

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